大家已经知道，1899年，希尔伯特第一次完成了欧几里得平面几何的公理化，其功绩永留人间。。

然而，历史并没有止步。1959年，数理逻辑学家塔尔斯基又进一步简化了希尔伯特的几何公理系统。请见“Tarski’s Axioms”.

1976年，塔尔斯基的得意弟子Keisler基于希尔伯特非阿基米德几何（超实数平面几何）创建了公理化的无穷小微积分（电子版）。

进入本世纪，学微积，用手机，蔚然成风，顺应发展浪潮，滚滚而去。

事实上，公理化无穷小微积分的背后，站立着两位世界数学巨人：希尔伯特与塔尔斯基。

袁萌  陈启清  3月16日

附件：塔尔斯基与希尔伯特几何公理系统之比较：

Comparison with Hilbert    Hilbert's axioms for plane geometry number 16, and include Transitivity of Congruence and a variant of the Axiom of Pasch. The only notion from intuitive geometry invoked in the remarks to Tarski's axioms is triangle. (Versions B and C of the Axiom of Euclid refer to '"circle" and "angle," respectively.) Hilbert's axioms also require "ray," "angle," and the notion of a triangle "including" an angle. In addition to betweenness and congruence, Hilbert's axioms require a primitive binary relation "on," linking a point and a line. The Axiom schema of Continuity plays a role similar to Hilbert's two axioms of Continuity. This schema is indispensable; Euclidean geometry in Tarski's (or equivalent) language cannot be finitely axiomatized as a first-order theory. Hilbert's axioms do not constitute a first-order theory because his continuity axioms require second-order logic.

The first four groups of axioms of Hilbert's axioms for plane geometry are bi-interpretable with Tarski's axioms minus continuity.